Atomic packing and size effect on the Hume-Rothery rule

Atomic packing problems have been widely investigated by physicists, mathematicians, material scientists as a long-standing scientific issue. While it is widely known that the monodisperse particles are closely packed into the face-centered cubic structure, the increase of polydispersity will suppress the crystallization. It is still unsolved when and how the crystalline packing collapses as the particle-size difference increases over a critical value. On the other hand, the atomic-size factor of the solution has gone deeply into the concept of applied physicists and physical metallurgy scientists since the proposal of Hume-Rothery rules in 1934. Although great efforts have been devoted to understanding this simple but important empirical rule of the atomic size effect, it remains a question without a complete answer even in binary alloy systems. Here, the two outstanding issues were solved in a rigorous geometric packing instability model even in the multicomponent system. The physical scenario presented here are helpful for understanding the close packing instability and the atomic-size effect in the Hume-Rothery rules.